Of graph, such one that it does not contain any directed cycle. A finite, acyclic digraph with no isolated vertices necessarily contain at least one source and at least one sink. See also directed acyclic graph (DAG for short) for more.

acyclic graph

A graph that contains no cycles.

adjacent

Two edges are adjacent if they have a node in common; two nodes are adjacent if they have an edge in common.

anti-edge

An edge that "is not there". More formally, for two vertices u{\displaystyle u} and v{\displaystyle v}, {u,v}{\displaystyle \{u,v\}} is an anti-edge in a graph G{\displaystyle G} if (u,v){\displaystyle (u,v)} is not an edge in G{\displaystyle G}. This means that there is either no edge between the two vertices or there is only an edge (v,u){\displaystyle (v,u)} from v{\displaystyle v} to u{\displaystyle u} if G{\displaystyle G} is directed.

anti-triangle

A set of three vertices none of which are connected.

arborescence

(also out-tree or branching) An oriented tree in which all vertices are reachable from a single vertex. Likewise, an in-tree is an oriented tree in which a single vertex is reachable from every other one.

adjacency matrix

In computers, a finite, directed or undirected graph (with n vertices, say) is often represented by its adjacency matrix: an n-by-nmatrix whose entry in row i and column j gives the number of edges from the i-th to the j-th vertex.

adjacent

Two vertices u and v are considered adjacent if an edge exists between them. We denote this by u ↓ v. In the above graph, vertices 1 and 2 are adjacent, but vertices 2 and 4 are not. The set of neighbors, called a (open) neighborhoodNG(v) for a vertex v in a graph G, consists of all vertices adjacent to v but not including v. When v is also included, it is called a closed neighborhood, denoted by NG[v]. When stated without any qualification, a neighborhood is assumed to be open. The subscript G is usually dropped when there is no danger of confusion. In the example graph, vertex 1 has two neighbors: vertices 2 and 5. For a simple graph, the number of neighbors that a vertex has coincides with its degree.

A circuit of n vertices, denoted by Cn, is usually assumed to be a simple cycle, or a simple circuit, meaning that every vertex is incident to exactly two edges. In the above graph (1, 5, 2, 1) is a simple cycle.

circumference

The length of a longest (simple) cycle, or infinity if the graph is acyclic.

Kn of order n, a simple graph with n vertices in which every vertex is adjacent to every other. The example graph is not complete. The complete graph on n vertices is often denoted by Kn. It has n(n-1)/2 edges (corresponding to all possible choices of pairs of vertices).

complete multipartite graph

A graph in which vertices are adjacent if and only if they belong to different partite sets.

component

A maximally connected subgraph.

connected

If some route exists from every node to every other, the graph is connected. Note that some graphs are not connected. A diagram of an unconnected graph may look like two or more unrelated diagrams, but all the nodes and edges shown are considered as one graph.

extends the concept of adjacency and is essentially a form (and measure) of concatenated adjacency.

connected graph

If it is possible to establish a path from any vertex to any other vertex of a graph, the graph is said to be connected.

k-connected

If it is always possible to establish a path from any vertex to every other one even after removing any k - 1 vertices, then the graph is said to be k-connected. Note that a graph is k-connected if and only if it contains k internally disjoint paths between any two vertices. The example graph above is connected (and therefore 1-connected), but not 2-connected.

connectivity

κ(G) of a graph G is the minimum number of vertices needed to disconnect G. By convention, Kn has connectivity n - 1; and a disconnected graph has connectivity 0.

crossing

A pair of intersecting edges.

crossing number

The minimum number of crossings that must appear when a graph is drawn on a plane is called the crossing number.

dG(v) of a vertex v in a graph G is the number of edges incident to v, with loops being counted twice. A vertex of degree 0 is an isolated vertex. A vertex of degree 1 is a leaf. In the example graph vertices 1 and 3 have a degree of 2, vertices 2,4 and 5 have a degree of 3 and vertex 6 has a degree of 1. If E is finite, then the total sum of vertex degrees is equal to twice the number of edges.

degree sequence

A list of degrees of a graph in non-increasing order (e.g. d1 ≥ d2 ≥ … ≥ dn). A sequence of non-increasing integers is realizable if it is a degree sequence of some graph.

(also cycle) An oriented simple cycle such that all arcs go the same direction, meaning all vertices have in- and out-degrees 1.

directed

A graph in which each edge symbolizes an ordered, non-transitive relationship between two nodes. Such edges are rendered with an arrowhead at one end of a line or arc.

directed graph

A graph whose edges are directed, possibly represented as ordered pairs of vertices; synonyms: digraph. Alternative models of graph exist; e.g., a graph may be thought as a Booleanbinary function over the set of vertices or as a square (0,1)-matrix.

disconnected graph

A graph that is not connected.

disconnecting set

A set of edges whose removal increases the number of components.

distance

dG(u, v) between two (not necessary distinct) vertices u and v in a graph G is the length of a shortest path between them. The subscript G is usually dropped when there is no danger of confusion. When u and v are identical, their distance is 0. When u and v are unreachable from each other, their distance is defined to be infinity ∞.

dominate

A vertex vdominates another vertex u if there is an arc from v to u. A vertex subset S is out-dominating if every vertex not in S is dominated by some vertex in S; and in-dominating if every vertex in S is dominated by some vertex not in S.

dominating set

Of a graph, a vertex subset whose closed neighborhood include all vertices of the graph. A vertex vdominates another vertex u if there is an edge from v to u. A vertex subset Vdominates another vertex subset U if every vertex in U is adjacent to some vertex in V. The minimum size of a dominating set is the domination number γ(G).

dual

A dual, or planar dual when the context needs to be clarified, G* of a plane graph G is a graph whose vertices represent the faces, including any outerface, of G and are adjacent in G* if and only if their corresponding faces are adjacent in G. The dual of a planar graph is always a planar pseudograph (e.g. consider the dual of a triangle). In the familiar case of a 3-connected simple planar graph G (isomorphic to a convex polyhedronP), the dual G* is also a 3-connected simple planar graph (and isomorphic to the dual polyhedron P*).

εG(v) of a vertex v in a graph G is the maximum distance from v to any other vertex.

edge

A set of two elements, drawn as a line connecting two vertices, called endvertices, or endpoints. An edge with endvertices x and y is denoted by xy without any symbol in between, so, do not write x⋅y. The edge set of G is usually denoted by E(G), or E when there is no danger of confusion.

edge, link, arc

Relationships represented in a graph. These are always rendered as straight or curved lines. The term "arc" may be misleading.

edge cut

The set of all edges having one endvertex in some proper vertex subset S and another endvertex in V(G)\S. Edges of K3 form a disconnecting set but not an edge cut. Any two edges of K3 form a minimal disconnecting set as well as an edge cut. An edge cut is necessarily a disconnecting set; and a minimal disconnecting set of an nonempty graph is necessarily an edge cut.

edgeless graph

(empty graph) A graph possibly with some vertices, but no edges. Or, it is a graph with no vertices and no edges.

k-edge-connected

Of a graph, such that if any subgraph formed by removing any k - 1 edges is still connected. The edge connectivity κ′(G) of a graph G is the minimum number of edges needed to disconnect G. One well-known result is that κ(G) ≤ κ′(G) ≤ δ(G).

embeddable

A graph is embeddable on a surface if its vertices and edges can be arranged on it without any crossing. The genus of a graph is the lowest genus of any surface on which the graph can embed.

embedding

G1=(V1,E1){\displaystyle G_{1}=(V_{1},E_{1})} of G2=(V2,E2){\displaystyle G_{2}=(V_{2},E_{2})} is an injection from V2{\displaystyle V_{2}} to V1{\displaystyle V_{1}} such that every edge in E2{\displaystyle E_{2}} corresponds to a path(disjoint from all other such paths) in G1{\displaystyle G_{1}}.

A path which passes through every edge (once and only once). If the starting and ending nodes are the same, it is an Euler cycle or an Euler circuit. If the starting and ending nodes are different, it is an Euler trail.

When a graph is drawn without any crossing, any cycle that surrounds a region without any edge reaching from the cycle inside to such region forms a face. Two faces on a plane graph are adjacent if they share a common edge.

factor

A spanning subgraph.

k-factor

A k-regular spanning subgraph. An 1-factor is a perfect matching. A partition of edges of a graph into k-factors is called a k-factorization. A k-factorable graph is a graph that admits a k-factorization.

Of a graph, the length of a shortest (simple) cycle in the graph, or infinity if the graph is acyclic.

graph, network

An abstraction of relationships among objects. Graphs consist exclusively of nodes and edges. All characteristics of a system are either eliminated or subsumed into these elements.

graph

A mathematical structure costiting of two types of elements, namely vertices and edges. Every edge has two endpoints in the set of vertices, ans is said to connect or join the two endpoints. The set of edges can thus be defined as a subset of the family of all two-element sets of vertices. Often, however, the set of vertices is considered as a set, and there is an incidence relation which maps each edge to the pair of vertices that are its endpoints.

The assignment of unique labels (usually natural numbers) to the edges and vertices of a graph. Graphs with labeled edges or vertices are known as labeled, those without as unlabeled. More specifically, graphs with labeled vertices only are vertex-labeled, those with labeled edges only are edge-labeled. (This usage is to distinguish between graphs with identifiable vertex or edge sets on the one hand, and isomorphism types or classes of graphs on the other.)

A path which passes through every node once and only once. If the starting and ending nodes are adjacent, it is a Hamiltonian cycle.

Hamiltonian path

A spanning path.

Hamiltonian connected graph

A graph that, given any pair of (distinct) vertices, contains a Hamiltonian path using them as endvertices.

H-free graf

A graph that does not contain H as an induced subgraph is said to be H-free.

head

The first vertex in the arc AKA directed edge. See also tail.

homomorphic

Likewise, a graph G is said to be homomorphic to a graph H if there is a mapping, called a homomorphism, from V(G) to V(H) such that if two vertices are adjacent in G then their corresponding vertices are adjacent in H.

hyperedge

An edge that is allowed to take on any number of vertices, possibly more than 2. A graph that allows any hyperedge is called a hypergraph. A simple graph can be considered a special case of the hypergraph, namely the 2-uniform hypergraph. However, when stated without any qualification, an edge is always assumed to consist of at most 2 vertices, and a graph is never confused with a hypergraph.

The number of edges entering a vertex. Is denoted as dΓ-(v). The degree dΓ(v) of a vertex v is equal to the sum of its out- and in- degrees. When the context is clear, the subscript Γ can be dropped. Maximum and minimum out degrees are denoted by Δ+(Γ) and δ+(Γ); and maximum and minimum in degrees, Δ-(Γ) and δ-(Γ).

in-neighborhood

(predecessor set) N-Γ(v) of a vertex v is the set of heads of arc going into v.

incident

An edgee connects two vertices; these two vertices are said to be incident to that edge, or, equivalently, that edge incident to those two vertices. All degree-related concepts have to do with adjacency or incidence.

independence number

α(G) of a graph G is the size of a largest independent set of G.

independent

In graph theory, the word independent usually carries the connotation of pairwise disjoint or mutually nonadjacent. In this sense, independence is a form of immediate nonadjacency.

independent set

A set of isolated vertices. Since the graph induced by any independent set is an empty graph, the two terms are usually used interchangeably. In the example above, vertices 1, 3, and 6 form an independent set; and 3, 5, and 6 form another one.

induced subgraph

A subgraph H of a graph G is said to be induced if, for any pair of vertices x and y of H, xy is an edge of H if and only if xy is an edge of G. In other words, H is an induced subgraph of G if it has the most edges that appear in G over the same vertex set. If H is chosen based on a vertex subset S of V(G), then H can be written as G[S] and is said to be induced by S.

infinite

A graph is infinite if it has infinitely many vertices or edges or both; otherwise the graph is finite. An infinite graph where every vertex has finite degree is called locally finite. When stated without any qualification, a graph is usually assumed to be finite.

initial vertex

A head.

internally disjoint

Two paths are internally disjoint (some people call it independent) if they do not have any vertex in common, except the first and last ones.

isolated vertex

A vertex not incident to any edges.

isomorphic

Two graphs G and H are said to be isomorphic, denoted by G ~ H, if there is a one-to-one correspondence, called an isomorphism, between the vertices of the graph such that two vertices are adjacent in G if and only if their corresponding vertices are adjacent in H.

A k-ary tree is a rooted tree in which every internal vertex has kchildren. An 1-ary tree is just a path. A 2-ary tree is also called a binary tree.

kernel

An independent out-dominating set. A digraph is kernel perfect if every induced sub-digraph has a kernel.

knot

In a directed graph, a collection of vertices and edges with the property that every vertex in the knot has outgoing edges, and all outgoing edges from vertices in the knot have other vertices in the knot as destinations. Thus it is impossible to leave the knot while following the directions of the edges.

The number of its edges. Cn has length n. A cycle, unlike a path, is not allowed to have length 0.

length of a path or walk

The number of edges that the path uses. Pn has length n - 1. Some people count multiple edges multiple times. In the example graph, (1, 2, 5, 1, 2, 3) is a path with length 5, and (5, 2, 1) is a simple path of length 2.

link

has two distinct endvertices.

digon

A cycle of length 2. In the example graph, (1, 2, 3, 4, 5, 1) is a cycle of length 5.

G2=(V2,E2){\displaystyle G_{2}=(V_{2},E_{2})} of G1=(V1,E1){\displaystyle G_{1}=(V_{1},E_{1})} is an injection from V2{\displaystyle V_{2}} to V1{\displaystyle V_{1}} such that every edge in E2{\displaystyle E_{2}} corresponds to a path(disjoint from all other such paths) in G1{\displaystyle G_{1}} such that every vertex in V1{\displaystyle V_{1}} is in one or more paths, or is part of the injection from V1{\displaystyle V_{1}} to V2{\displaystyle V_{2}}. This can alternatively be phrased in terms of contractions, which are operations which collapse a path and all vertices on it into a single edge (see edge contraction).

multiple

A set of arcs are multiple, or parallel, if they share the same head and the same tail. A pair of arcs are anti-parallel if one's head/tail is the other's tail/head.

multiple edge

An edge such that there is another edge with the same endvertices; antonyms: simple edge. The multiplicity of an edge is the number of multiple edges sharing the same endvertices; the multiplicity of a graph, the maximum multiplicity of its edges.

multigraph

A graph that has multiple edges, but no loops.

multipartite graph

A graph that can be decomposed into an unspecific number of partite sets but not fewer.

A weighted graph, possibly directed or undirected, possibly containing special vertices (nodes), such as source or sink.

null graph

The graph with no vertices and no edges. Or, it is a graph with no edges and any number n{\displaystyle n} of vertices, in which case it may be called the null graph on n{\displaystyle n} vertices. (There is no consistency at all in the literature.)

An assignment of directions to the edges of an undirected or partially directed graph. When stated without any qualification, it is usually assumed that all undirected edges are replaced by a directed one in an orientation. Also, the underlying graph is usually assumed to be undirected and simple.

oriented graph

A graph that contains only arcs. When stated without any qualification, a graph is almost always assumed to be undirected. Also, a digraph is usually assumed to contain no undirected edges.

out degree

The number of edges leaving a vertex. Is denoted dΓ+(v), given a digraph Γ. See also in degree.

out-neighborhood

(successor set) N+Γ(v) of a vertex v is the set of tails of arcs going from v.

outer face

Furthermore, since we can establish a sense of "inside" and "outside" on a plane, we can identify an "outermost" region that contains the entire graph if the graph does not cover the entire plane. Such outermost region is called an outer face.

outerplanar graph

A graph that can be drawn in the planar fashion such that its vertices are all adjacent to the outer face. See also outerplane graph.

outerplane graph

A graph that is drawn in the planar fashion such that its vertices are all adjacent to the outer face. See also outerplanar graph.

A graph that contains cycles of every possible length (from 3 to the order of the graph).

k-partite graph

A graph that can be decomposed into k partite sets but not fewer.

partite set

A graph can be decomposed into independent sets in the sense that the entire vertex set of the graph can be partitioned into pairwise disjoint independent subsets. Such independent subsets are called partite sets, or simply parts.

path

A directed path, or just a path when the context is clear, is an oriented simple path such that all arcs go the same direction, meaning all internal vertices have in- and out-degrees 1.

path

A route that does not pass any edge more than once. If the path does not pass any node more than once, it is a simple path.

path

In a graph was what is now usually known as an open walk. Nowadays, when stated without any qualification, a path of n vertices, denoted by Pn (but some write Pn-1), is usually defined to be a simple path or a simple trail in the old sense, meaning that every vertex is incident to at most two edges.

The minimum eccentricity over all vertices in a graph. It is denoted rad(G) of a graph G. When there are two components in G, diam(G) and rad(G) defined to be infinity ∞.

reachable

Of a vertex v from a given vertex u in a directed graph, such that there is a directed path that starts from u and ends at v.

regular

A graph in which each node has the same degree.

regular graph

A graph in which every vertex has the same degree.

k-regular graph

A graph in which every vertex has degree k. A 0-regular graph is an independent set. A 1-regular graph is a matching. A 2-regular graph is a vertex disjoint union of cycles. A 3-regular graph is said to be cubic, or trivalent.

route

A sequence of edges and nodes from one node to another. Any given edge or node might be used more than once.

A spanning subgraph that is a tree. Every graph has a spanning forest. But only a connected graph has a spanning tree.

star

A special kind of tree called star is K1,k; see also claw.

strongly connected

A digraph is strongly connected if every vertex is reachable from every other following the directions of the arcs. On the contrary, a diagraph is weakly connected if its underlying undirected graph is connected. A weakly connected graph can be thought of as a digraph in which every vertex is "reachable" from every other but not necessarily following the directions of the arcs. A strong orientation is an orientation that produces a strongly connected digraph.

strongly connected component

Informally, a subgraph where all nodes in the subgraph are reachable by all other nodes in the subgraph.

subtree

of the graph G is a subgraph that is a tree.

subgraph

Of a graph G is a graph whose vertex and edge sets are subsets of those of G. In the other direction, a supergraph of a graph G is a graph that contains G as a subgraph. We say a graph Gcontains another graph H if some subgraph of G is H or is isomorphic to H (depending on the needs of the situation).

semiregular

A k-partite graph is semiregular if each of its partite set has a uniform degree

spanning matching

A matching that covers all vertices of a graph.

stable set

An independent set.

staset

A stable set.

strongly regular graph

A regular graph such that any adjacent vertices have the same number of common neighbors as other adjacent pairs and that any nonadjacent vertices have the same number of common neighbors as other nonadjacent pairs.

The union of three internally disjoint (simple) paths that have the same two distinct endvertices. A theta0 graph has seven vertices which can be arranged as the vertices of a regular hexagon plus an additional vertex in the center. The eight edges are the perimeter of the hexagon plus one diameter.

thickness

The minimum number of planar graphs needed to cover a graph is the thickness of the graph.

totally disconnected graph

A graph is totally disconnected if there is no path connecting any pair of vertices. This is just another name to describe an empty graph or independent set.

tournament

A digraph in which each pair of vertices is connected by exactly one arc. In other words, it is an oriented complete graph.

traceable graph

A graph that contains a Hamiltonian path.

traceable path

A spanning path.

trail

A walk in which all the edges are distinct. (A closed trail has been called a tour or circuit, but these are uncommon and the latter is usually reserved for a regular subgraph of degree two.)

A connected acyclic simple graph. A vertex of degree 1 is called a leaf, or pendant vertex. An edge incident to a leaf is an leaf edge, or pendant edge. (Some people define a leaf edge as a leaf and then define a leaf vertex on top of it. These two sets of definitions are often used interchangeably.) A non-leaf vertex is an internal vertex. Sometimes, one vertex of the tree is distinguished, and called the root. A rooted tree is a tree with a root. Rooted trees are often treated as directed acyclic graphs with the edges pointing away from the root. Trees are commonly used as data structures in computer science (see tree data structure).

An alternating sequence of vertices and edges, beginning and ending with a vertex, in which each vertex is incident to the two edges that precede and follow it in the sequence, and the vertices that precede and follow an edge are the endvertices of that edge. A walk is closed if its first and last vertices are the same, and open if they are different.

weighted

Weighted edges symbolize relationships between nodes which are considered to have some value, for instance, distance or lag time. Such edges are usually annotated by a number or letter placed beside the edge. Weighted nodes also have some value; this must be distinguished from identification.

weighted graph

A graph that associates a label (weight) with every edge in the graph. Weights are usually real numbers. They may be restricted to rational numbers or integers. Certain algorithms require further restrictions on weights; for instance, the Dijkstra algorithm works properly only for positive weights.

weight of a subgraph

The sum of the weights of the edges of the subgraph.

Wiener index of a vertex

v in a graph G, denoted by WG(v) is the sum of distances between v and all others.

Wiener index of a graph

G, denoted by W(G), is the sum of distances over all pairs of vertices.